METRIZATION

Many problems that analysts study are not primarily concerned with
a single object such as a function, a measure, or an operator, but they deal
instead with large classes of such objects. Most of the interesting classes
that occur in this way turn out to be vector spaces, either with real scalars
or with complex ones. Since limit processes play a role in every analytic
problem (explicitly or implicitly), it should be no surprise that these vector
spaces are supplied with metrics, or at least with topologies, that bear some
natural relation to the objects of which the spaces are made up. The simplest
and most important way of doing this is to introduce a norm. The
resulting structure (defined below) is called a normed vector space, or a
normed linear space, or simply a normed space.
Throughout this book, the term vector space will refer to a vector
space over the complex field ({ or over the real field R.Many problems that analysts study are not primarily concerned with
a single object such as a function, a measure, or an operator, but they deal
instead with large classes of such objects. Most of the interesting classes
that occur in this way turn out to be vector spaces, either with real scalars
or with complex ones. Since limit processes play a role in every analytic
problem (explicitly or implicitly), it should be no surprise that these vector
spaces are supplied with metrics, or at least with topologies, that bear some
natural relation to the objects of which the spaces are made up. The simplest
and most important way of doing this is to introduce a norm. The
resulting structure (defined below) is called a normed vector space, or a
normed linear space, or simply a normed space.
Throughout this book, the term vector space will refer to a vector
space over the complex field ({ or over the real field R.